Presentation on theme: "Statistical Mechanics and Evolutionary Theory"— Presentation transcript:
1 Statistical Mechanics and Evolutionary Theory Lloyd DemetriusHarvard University, Cambridge, Mass., USAAndMax Planck Institute, Berlin, Germany
2 Evolutionary changes in morphological complexity Ecological time scale (Single evolving lineage)Increases and decreases in adult body sizeGeological time scale (Phyletic lineages)Increases in mean body sizeGeological time scale (Clades)Increases in maximum body size
7 Increase in maximum body size over the history of life
8 ProblemWhat is the evolutionary basis for the changes in body size over evolutionary time ?
9 Darwinian argumentIndividuals differ in terms of their morphology, behavior and other phenotypic characteristics (variation)Different phenotypes are characterized by differences in the acquisition and transformation of resources (natural selection)There exists a correlation between the characteristics of parents and their offspring (heredity)
10 Darwinian fitnessThe efficiency with which organisms transform resources into net offspring production
11 Levels of biological organization Populational: Changes in the phenotypic composition of a population by a natural selectionPhyletic lineage: Changes in the species composition of a lineage by speciation and background extinctionClade: Changes in the species composition of a clade by speciation and mass extinction
12 If W2 > W1 : then X2 replaces X1 Darwinian modelOrganic diversity and changes in complexity can be explained in terms of the following tenetSelection tenetResident type X1 ; Fitness W1Variant type X2 ; Fitness W2If W2 > W1 : then X2 replaces X1FitnessThe efficiency to transform respurces into net-offspring productionX1X2
13 Darwin‘s theoryEvolutionary Principle: Evolution by natural selection results in an increase in fitnessExplanatory PowerVariation in life history, body size, life span within and between speciesThe adaptation of species to their habitatThe changes in morphological complexity over time
14 ProblemCan Darwin‘s argument be translated into an analytical theory which will explain:The diversity of species in space and timeThe adaptation of species to their environmentThe increase in complexity within lineages
15 Does there exist a demographic characterization of fitness which will predict the outcome of competition between variants and incumbents in a population of organisms ?
16 Characterizations of Darwinian Fitness Malthusian parameter (1930) Fisher‘s theoryEvolutionary entropy (1974) Directionality theory
17 The theory of evolution by natural selection is the doctrine of Malthus applied to plantsand animals.Darwin (1859)
18 Population described by d age-classes Demographic modelPopulation described by d age-classesbi = Probability of surviving from age-class (i) to age-class(i+1)mi = Mean number of offspring produced by individual inage-class (i)lj = b1,b2,...,bj-1 = Survivorship to age-class (j)Vj = lj mj = Net-reproduction at age j
19 Malthusian parameter as Darwinian fitness Matrix Representation of GraphCharacterization of r :
20 Growth rate r characterizes Darwinian Fitness: Fisher‘s TheoryGrowth rate r characterizes Darwinian Fitness:Malthusian Principle: r predicts the outcome of competition between variant and incumbent typesrr*rX*XX*Xr*
21 Fisher‘s evolutionary theory Population growth rateMean FitnessFisher‘s principle: Evolution by natural selection results in an increase in the mean malthusian parameter
22 The Malthusian Parameter as Darwinian Fitness Critique Computational studies: In Competition between mutants and the resident population the growth rate is not always a good predictor of invasion successEmpirical studies: Invasion success is highly correlated with body size and is contingent on the resource constraints
23 Darwin‘s theory of evolution by natural selection is the doctrine of Gibbs, Boltzmann and Clasius applied to plants and animals.Directionality theory(1974)
24 Directionality theory Evolutionary entropy, S , characterizes Darwinian Fitness
25 Evolutionary principles Evolutionary dynamics within a single evolving lineage(Mutation and Selection)Directionality Principle for EntropyLimited Resources: Evolution increases entropyVariable Resources: Evolution decreases entropyEvolutionary dynamics within a taxon(Speciation and Extinction)Fundamental Theorem of EvolutionThe rate of change of mean entropy is equal to the variance in entropyMean entropy increases over geological timeEvolutionary dynamics within a a clade( Speciation, background and mass extinction )Secondary Theorem of EvolutionThe upper entropic limit of species in a clade increases as the claded replaces another over geological time
26 Organization The origin of evolutionary entropy: Its demographic basis The directionality principles for evolution:Their mathematical basisImplications of directionality theory for the study ofLife history evolutionEvolution of body sizeEvolution of senescence
27 Origin of evolutionary entropy Demographic model Microstates:Population growth rate:
28 Biological networks Macrostates from microstates PAnn. App. Prob. (1974)P3.
29 Demographic networks Macrostates from microstates Entropy:Reproductive potential:Generation time:
30 Properties of entropy Measure of uncertainty Measure of diversity Measure of robustness
31 Uncertainty measureUncertainty in the age of the mother a randomly chosen newbornpj Probability that the mother of a randomly chosen newborn belongs to age class (j)
32 Robustness Genealogies: Set of paths of the graph Path: Matrix associated with the graph132d...
33 Robustness Theorem: Annals. App. Prob.(1994) Prob. that the sample meanTheorem:Annals. App. Prob.(1994)
34 Reproduction potential and resource constraints Proposition: In Populations in dynamical equilibrium with resource conditionsE<0: Constant resourceE>0: Variable resource
35 The Entropic Selection Principle Entropy as darwinian fitnessCompetition betweem variant and incumbent is a stochastic process determined by entropy (S) and contingent on the resource constraints (E)Limited resources: (E<0)Mutants with increased entropy have increased robustness and will prevail (a.s)Variable resources: (E>0)Large population size:Mutants with decreased entropy will have decreased robustness and will prevail (a.s)Small population size:The outcome of competition will be a stochastic process described by probabilities contigent on population size
36 Invasion dynamics Evolutionary entropy predicts the outcome of competition Limited ResourcesXX*SS*XX*S*SVariable ResourcesXX*SS*
37 Predictions of directionality theory Based on the entropic principes of selection we predict the evolutionary changes at three different levels of biological organization.Single evolving lineage – Mutation and selectionAggregate of phyletic lineages – Speciation and background extinctionAn ensemble of clades – Speciation and mass extinction
38 Evolutionary dynamics within an evolving lineage Long run changes in entropy as one population type replaces another under mutation and natural selectionEquilibrium species: Species subject to limited resource conditionsOpportunistic species: Species subject to variable resource conditionsEvolutionary principles:Entropy increases in equilibrium speciesEntropy decreases in opportunistic species
39 Evolutionary dynamics within a taxon Long run changes in mean entropy as one phyletic lineage replaces another under speciation and background extinction.The rate of change in mean entropy is equal to variance in entropyMean entropy increases
40 Evolutionary dynamics within a clade Long run changes in maximum entropy as one clade replaces another under mass extinctionThe upper entropic limit increases as one clade replaces another over geological time.
41 Main tenets of the evolutionary process Evolutionary dynamics within a single evolving lineageEquilibrium species: Entropy increasesOpportunistic species: Entropy decreasesEvolutionary dynamics within a taxonThe rate of change of mean entropy is equal to the variance in entropyEvolutionary dynamics within a cladeThe upper entropic limit increases as one clade replaces another
42 Implications of the evolutionary tenets Evolution of life historyEvolution of body sizeEvolution of senescence
43 Body size and physiological time Allometric relationsBody size and physiological timePhysica A.(2003)Physiological time, Body sizePhysiological timeCycle time of metabolic processesGeneration timeLife span
45 The evolution and distribution of species body size Relation between entropy Sand body size W
46 Empirical study Relation between entropy and body size
47 Directionality theory predicts evolutionary changes in body size Changes in body size within a single evolving lineageLimited resource conditionsIncrease in body sizeVariable resource conditionsDecrease in body size
48 Changes in body size within the equid lineages Increase in body size : North AmericaDecrease in body size : Europe
49 Directionality theory predicts evolutionary change in body size within a taxon The rate of change of the mean body size of species within a phyletic lineage is equal to the species variance in body sizeMean body size increases within a taxon( Cope‘s Rule )
50 Increase in mean body size within the equid taxon
51 Evolutionary changes in the upper limit of bodysize The upper limit of body size increases as one clade replaces another over geological time.
55 The evolution of senescence Directionality theory explains variation in the rate of aging between equilibrium and opportunistic species.Proposition: The intensity of natural selection is a convex function of age
57 Conclusion Darwinian Fitness is characterized by evolutionary entropy Diversity of species and evolutionary change in complexity can be described in terms of the following tenets:Population level:Equilibrium species: Entropy increasesOpportunistic species: Entropy decreasesPhyletic level:Mean entropy increasesClade:The upper entropic limit increases
58 Relation between thermodynamic variables and evolutionary parameters Free energy,Thermodynamic entropy,Temperature,Mean energy,Evolutionary parametersGrowth rate,Demographic rate, Reciprocal generation time,Reproductive potential
59 Relation between thermodynamic principles and evolutionary principles Thermodynamic entropy:Diversity of energy distributionDemographic entropy:Diversity of energy flowThe entropic principle for evolution is a non—equilibrium analogue of the entropic principle for physical systems.
60 Relation between thermodynamic principles and evolutionary principles Thermodynamic entropy:Demographic entropy:Analytic relation between generation time, and Temeprature :Theorem: The entropc principle for thermodynamic systems is the limit of the entropic principle for evolutionary processes.